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Formation energies of point defects at finite temperatures
The dominant entropy contribution affecting defect concentrations is configurational entropy. Other contributions such as harmonic and anharmonic lattice vibrations are second order effects and computationally expensive to calculate. Therefore, such contributions have been rarely considered in defect investigations. However, to achieve the next accuracy level in defect calculations and thus significantly improve the agreement with experiment, the inclusion of these contributions is critical. In this paper, we present the methods needed to compute highly accurate free energies of point defects from first principles. We demonstrate how to include all relevant free energy contributions up to the melting point. The focus will be on nonmagnetic metals and point defects in the dilute limit. We consider all relevant excitation mechanisms: electronic excitations and ionic vibrations both in the quasiharmonic approximation and explicitly including anharmonicity (i.e., phonon–phonon interaction). Since computing such interactions requires to sample large parts of the phase space, straightforward ab initio based simulations (such as molecular dynamics) are in most cases out of reach even on supercomputers. To overcome this difficulty, a recently developed hierarchical scheme will be presented which allows to coarse grain the configuration space and thus to efficiently calculate anharmonic contributions to defect formation. We discuss the performance and accuracy of the developed methodology for the example of vacancies in aluminum. An important insight is that the entropy of vacancy formation is significantly affected by anharmonicity. We further show that the inclusion of all the aforementioned excitation mechanisms is critical to guarantee an accurate description of thermodynamic properties up to the melting point.